Some world have a strong resistance come using clinical notation*. Practically every time i teach an introductory science class, I have one or 2 students with solid math skills who insist on doing all of their calculations in typical notation. Doing this invariably outcomes in failure that lead to lost point out on exams and also homework.

You are watching: 10 to the 7th power in standard form

**Video Overview**

Making sense of scientific notation does take it a little bit of effort, however once learned, it becomes an effective method to conserve time and avoid mistakes once working v extremely huge and little numbers. This is especially helpful when making use of calculators or computer systems for used work. That is much less complicated to avoid mistakes when writing 4.6x10-14 as compared to composing 0.000000000000046.

*Numbers written in scientific notation are much easier to read and to work with*

Converting a number from standard to scientific notation entails moving the decimal ar until there is one digit to the left the the decimal, climate multiplying that value by ten elevated to the exact same power as the variety of places moved. 300 created in clinical notation is 3x102; spoke together "Three time ten to the two." Ten elevated to the strength of two shows that to convert the value earlier to traditional notation, the decimal needs to relocate two places. The authorize of the exponent* relies on the direction the decimal moves.

*Move the slider come compare values in standard and also scientific notation*

All values written in clinical notation have the exact same structure: The number itself, followed by a multiplication symbol, then the number 10 elevated to a power.

The number is dubbed the coefficient. The strength 10 is elevated to is called the exponent. The coefficient can be any kind of real number. The exponent need to be an integer, back it have the right to be negative.

*The parts of the notation structure have actually names*

A confident exponent way the number is greater than one, and also the decimal move to the ideal to convert from scientific to traditional notation. 3.0 x102 becomes 300.

A an unfavorable exponent method the number is less than one, and the decimal moves to the left to transform from clinical to traditional notation. 3.0 x10-2 becomes 0.03.

Moving the decimal the same number of places 10 is raised to works since moving the decimal one ar is the exact same as dividing or multiplying the number by 10:

Step 1 - divide 300 by 10: 300 ÷ 10 = 30Step 2 - division the result of step 1 through 10 again: 30 x 101 ÷ 10 = 3Since we had actually to division by 10 two times to obtain from 300 come 3, the exponent for the scientific notation type is 2. This reflects that the initial number deserve to be recreated by multiply 3 by 10 two times.

Therefore, 3.0 x 102, 3.0 x 10 x 10 and 300 all represent the same value.

The same pattern functions regardless the the size of the number. 2,300,000,000,000 can be written as 2.3 x 1012 which claims it is 2.3 multiply by 10 12 times.

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*Pattern functions for huge numbers too*

Values that are much less than one job-related the exact same way. The difference is that converting from typical to scientific notation requires multiplying through 10, no dividing. Multiply by 10 moves the decimal to the right. To compose the number 0.0003 in clinical notation, 0.0003 is repeatedly multiplied by 10 till the an initial non-zero number (in this instance a 3) is simply to the left the the decimal place:

Step 1 - main point by 10: 0.0003 x 10 = 0.003Step 2 - main point by 10 again: 0.003 x 10-1 x 10 = 0.03Step 3 - multiply by 10 again: 0.03 x 10-2 x 10 = 0.3Step 4 - main point by 10 again: 0.3 x 10-3 x 10 = 3This conversion take it 4 procedures so the number is multiplied by 10 elevated to the negative of the number so procedures indicating that to getting back to the original number requires dividing by 10 4 times. The exponent is an adverse because the initial number is less than one.

In enhancement to making it less complicated to express big and small numbers, clinical notation makes it easier to compare values. Writing 2 values as:

0.000000035 and also 0.0000000098

makes it difficult to check out at a glance which one is enlarge while writing them as

3.5 x10-8 and 9.8 x10-9

makes it much clearer the the first one is larger.

The many common method to write worths in scientific notation is v the x10exponent structure. With the introduction of computers and calculators, the letter E replaces "x10" so the 3200 can be created as 3200, 3.2E3, or 3.2 x103.

Try the clinical notation practice difficulties to test your understanding of the ideas covered by this illustration.